We show that the existence of a ``good'' coupling w.r.t. Hamming distance for any local Markov chain on a discrete product space implies rapid mixing of the Glauber dynamics in a blackbox fashion. More specifically, we only require the expected distance between successive iterates under the coupling to be summable, as opposed to being one-step contractive in the worst case. Combined with recent local-to-global arguments \cite{CLV21}, we establish asymptotically optimal lower bounds on the standard and modified log-Sobolev constants for the Glauber dynamics for sampling from spin systems on bounded-degree graphs when a curvature condition \cite{Oll09} is satisfied. To achieve this, we use Stein's method for Markov chains \cite{BN19, RR19} to show that a ``good'' coupling for a local Markov chain yields strong bounds on the spectral independence of the distribution in the sense of \cite{ALO20}. Our primary application is to sampling proper list-colorings on bounded-degree graphs. In particular, combining the coupling for the flip dynamics given by \cite{Vig00, CDMPP19} with our techniques, we show optimal $O(n\log n)$ mixing for the Glauber dynamics for sampling proper list-colorings on any bounded-degree graph with maximum degree $\Delta$ whenever the size of the color lists are at least $\left(\frac{11}{6} - \epsilon\right)\Delta$, where $\epsilon \approx 10^{-5}$ is small constant. While $O(n^{2})$ mixing was already known before, our approach additionally yields Chernoff-type concentration bounds for Hamming Lipschitz functions in this regime, which was not known before. Our approach is markedly different from prior works establishing spectral independence for spin systems using spatial mixing \cite{ALO20, CLV20, CGSV20, FGYZ20}, which crucially is still open in this regime for proper list-colorings.

中文翻译：

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从耦合到光谱独立，以及黑匣子与下移步法的比较

我们表明，离散产品空间上任何局部马尔可夫链的``汉明距离''耦合均良好，这意味着以黑匣子的方式快速混合了格劳伯动力学。更具体地说，我们只要求在耦合下连续迭代之间的预期距离是可累加的，而不是在最坏的情况下是一步收缩。结合最近的局部到全局参数\ cite {CLV21}，我们在标准曲面和修改后的log-Sobolev常数上建立了Glauber动力学的渐近最优下界，以便在曲率条件\ cite时从有界图上的自旋系统采样{Oll09}很满意。为了达到这个目的，我们对马尔可夫链\ cite {BN19，RR19}使用Stein方法 局部马尔可夫链的耦合在\ cite {ALO20}的意义上对分布的光谱独立性产生了严格的界限。我们的主要应用是在有界图上对适当的列表颜色进行采样。特别是，将\ cite {Vig00，CDMPP19}给出的翻转动力学耦合与我们的技术结合起来，我们展示了针对Glauber动力学的最佳$ O（n \ log n）$混合，可以在任何有界的像素上采样适当的列表颜色只要颜色列表的大小至少为$ \ left（\ frac {11} {6}-\ epsilon \ right）\ Delta $，其中$ \ epsilon \ approx 10 ^ { -5} $是小常数。尽管以前已经知道$ O（n ^ {2}）$混合，但是我们的方法还可以为这种形式的Hamming Lipschitz函数生成Chernoff型浓度边界，这是以前未知的。